In cryptocurrencies like Bitcoin, private keys play a crucial role in ensuring security and access to funds. However, in some cases, private keys can be lost or damaged, making it impossible to access the wallet. One method to restore access is to use repeated generations of private keys based on the Birthday Paradox. In this article, we will look at the mathematical basis of this method and its application to restoring private keys in Bitcoin wallets.

Birthday Paradox

The birthday paradox is a statistical phenomenon that shows that in a group of 23 people, the probability that at least two of them will have the same birthday is greater than 50%.[4] The paradox is based on the idea that the number of possible coincidences increases exponentially with sample size.

Applying the Birthday Paradox to Private Key Recovery

In the context of cryptocurrencies, private keys are 64-character sequences consisting of letters from “a” to “f” and numbers from “0” to “9”[3]. This gives $$16^{64}$$ possible keys. However, if we generate keys randomly, we may encounter a situation where two different keys yield the same wallet address, which is a consequence of a hash function collision.

Mathematical basis for private key recovery

To recover a private key through repeated generations, you can use the following approach:

1. Key generation : Private keys are randomly generated and the corresponding wallet addresses are calculated.
2. Match check : Checks if the generated keys match the target wallet address.
3. Key Recovery : If a match is found, the corresponding private key can be used to restore access to the wallet.

Mathematical model

Let $$N$$ be the number of generated keys and $$P$$ be the probability of finding a match. The probability $$P$$ can be estimated as:

$$
P = 1 — \left(1 — \frac{1}{16^{64}}\right)^N
$$

This formula shows that as $$N$$ increases, the probability of finding a match increases.

Problems and limitations

1. Computational complexity : Generating and verifying a large number of keys requires significant computational resources.
2. Probability of success : The probability of finding a match remains low, even for large $$N$$, due to the huge space of possible keys.
3. Security : Using this method can be unsafe as it is based on random generation and does not guarantee finding the correct key.

Conclusion

Recovering private keys through repeated generations based on the birthday paradox is theoretically possible, but impractical due to the computational complexity and low success rate. More effective methods of recovering access to a wallet may be to use backups or recover through other available data, such as redundantly encoded QR codes[2]. It is important to always store private keys in a safe place to avoid the need for recovery.

Citations:
[1] https://habr.com/ru/articles/683802/
[2] https://pikabu.ru/story/vosstanovlenie_povrezhdennogo_zakryitogo_klyucha_btc_10963903
[3] https://dzen.ru/a/ZTvInA6yFiSidbZS
[4] https://cryptodeep.ru/quantum-computer-qianshi/
[5] https://pikabu.ru/@CryptoDeepTech
[6] https://www.youtube.com/watch?v=XyM6SEi2h7I
[7] https://pikabu.ru/tag/Free%20bitcoin,YouTube/best?page=2&page=50
[8] https://ikiacademy.org/uploads/0-1-202208211661083057.pdf